具有最小高斯表面积的对称凸集
Symmetric Convex Sets with Minimal Gaussian Surface Area
摘要：
设$Ω\subset\mathbb{R}^{n+1}$在满足$Ω=-Ω$的所有集合中具有最小的高斯表面积，且具有固定的高斯体积。令$A=A_{x}$为$\partialΩ$在$x$处的第二基本形式，即$A$为单位法向量在$x\in\partialΩ$处的一阶偏导数矩阵。对于任何$x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}$，设为$γ_{n}(x)=(2π)^{-n/2}e^{-(x_{1}^{2}+\cdots+x_{n+1}^{2})/2}$。令$\|A\|^{2}$是$A$的项的平方和，并且令$\|A\|_{2\to 2}$表示$A$的$\ell_{2}$算子范数。证明了若$Ω$或$Ω^{c}$是凸的，且若$$\int_{\partialΩ}(\|A_{x}\|^{2}-1)γ_{n}(x)dx>0\qquad\mbox{or}\qquad \int_{\partialΩ}\Big(\|A_{x}\|^{2}-1+2\sup_{y\in\partialΩ}\|A_{y}\|_{2\to 2}^{2}\Big)γ_{n}(x)dx<0,$$中有一个是凸的，则$\partialΩ$必是圆柱.也就是说，除了$\|A\|^{2}$的平均值略小于$1$的情况外，我们解决了2001年Barthe问题的凸情况。主要工具是高斯极小曲面的Colding-Minicozzi理论，它研究了与曲面$\partialΩ$相关的Ornstein-Uhlenbeck型算子$L= Δ-\langle x,\nabla \rangle+\|A\|^{2}+1$的本征函数。一个关键的新成分是在高斯表面积的第二变分公式中使用随机选择的2次多项式。我们的实际结果比上面的陈述更一般一些。另外，我们的一些结果在没有凸性假设的情况下也成立。
英文摘要：
Let $Ω\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $Ω=-Ω$ with fixed Gaussian volume. Let $A=A_{x}$ be the second fundamental form of $\partialΩ$ at $x$, i.e. $A$ is the matrix of first order partial derivatives of the unit normal vector at $x\in\partialΩ$. For any $x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}$, let $γ_{n}(x)=(2π)^{-n/2}e^{-(x_{1}^{2}+\cdots+x_{n+1}^{2})/2}$. Let $\|A\|^{2}$ be the sum of the squares of the entries of $A$, and let $\|A\|_{2\to 2}$ denote the $\ell_{2}$ operator norm of $A$. It is shown that if $Ω$ or $Ω^{c}$ is convex, and if either $$\int_{\partialΩ}(\|A_{x}\|^{2}-1)γ_{n}(x)dx>0\qquad\mbox{or}\qquad \int_{\partialΩ}\Big(\|A_{x}\|^{2}-1+2\sup_{y\in\partialΩ}\|A_{y}\|_{2\to 2}^{2}\Big)γ_{n}(x)dx<0,$$ then $\partialΩ$ must be a round cylinder. That is, except for the case that the average value of $\|A\|^{2}$ is slightly less than $1$, we resolve the convex case of a question of Barthe from 2001. The main tool is the Colding-Minicozzi theory for Gaussian minimal surfaces, which studies eigenfunctions of the Ornstein-Uhlenbeck type operator $L= Δ-\langle x,\nabla \rangle+\|A\|^{2}+1$ associated to the surface $\partialΩ$. A key new ingredient is the use of a randomly chosen degree 2 polynomial in the second variation formula for the Gaussian surface area. Our actual results are a bit more general than the above statement. Also, some of our results hold without the assumption of convexity.
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